MNRAS 461, 4406–4418 (2016) doi:10.1093/mnras/stw1577 Advance Access publication 2016 July 4

On the orbital evolution of a pair of giant in mean motion resonance

Q. Andre´1,2‹ and J. C. B. Papaloizou1 1DAMTP, University of Cambridge, Wilberforce Road, Cambridge CB3 OWA, UK 2Departement´ de Physique, ENS Cachan, Universite´ Paris-Saclay, 61 Avenue du President´ Wilson, F-94230 Cachan, France

Accepted 2016 June 30. Received 2016 June 30; in original form 2015 July 9

ABSTRACT Pairs of extrasolar giant planets in a mean motion commensurability are common with 2:1 Downloaded from resonance occurring most frequently. Disc– interaction provides a mechanism for their origin. However, the time-scale on which this could operate in particular cases is unclear. We perform 2D and 3D numerical simulations of pairs of giant planets in a protoplanetary disc as they form and maintain a mean motion commensurability. We consider systems with http://mnras.oxfordjournals.org/ current parameters similar to those of HD 155358, 24 Sextantis and HD 60532, and disc models of varying , decreasing mass corresponding to increasing age. For the lowest mass discs, systems with planets in the Jovian mass range migrate inwards maintaining a 2:1 commensurability. Systems with the inner planet currently at around 1 au from the central could have originated at a few au and migrated inwards on a time-scale comparable to protoplanetary disc lifetimes. Systems of larger mass planets such as HD 60532 attain 3:1 resonance as observed. For a given mass accretion rate, results are insensitive to the disc model for the range of viscosity prescriptions adopted, there being good agreement between 2D and 3D simulations. However, in a higher mass disc a pair of Jovian mass planets passes through at University of Cambridge on August 10, 2016 2:1 resonance before attaining a temporary phase lasting a few thousand orbits in an unstable 5:3 resonance prior to undergoing a scattering. Thus, finding systems in this commensurability is unlikely. Key words: accretion, accretion discs – hydrodynamics – methods: numerical – planet–disc interactions – protoplanetary discs – planetary systems.

figurations in situ is expected to be small (e.g. Beauge, Ferraz-Mello 1 INTRODUCTION & Michtchenko 2012). Pairs of extrasolar giant planets in a mean motion commensurability Disc–planet interaction can produce the required evolution of the are a common occurrence. It has been estimated that sixth of multi- semi-major axes. This may result in convergent migration leading planet systems detected by the technique are in or to the formation of a commensurability (see Baruteau et al. 2014 close to a 2:1 commensurability (Wright et al. 2011). Parameters for and references therein). Accordingly, understanding the observed some cases of interest that are considered in this paper are shown configuration of such systems has the potential for either revealing in Table 1 (for additional examples see e.g. Emelyanenko 2012). how disc–planet interactions may have operated or for ruling them In addition, there are two known systems in 3:2 resonance (Correia out. et al. 2009;Reinetal.2012; Robertson et al. 2012b) and one in a Previous numerical studies of commensurabilities forming and 4:3 resonance (Johnson et al. 2011;Reinetal.2012) consistent with evolving as a result of disc–planet interactions have focused on the systems in 2:1 resonance being the most commonly observed systems such as GJ 876, HD 45364 and HD 6805 interacting with commensurability. disc modelled with either constant kinematic viscosity or with the The existence of these resonant systems indicates that dissipative α-viscosity parameter of Shakura & Sunyaev (1973) taken to be con- mechanisms that result in changes to planet semi-major axes that stant (for a review see Baruteau et al. 2014 and references therein). produce related changes to period ratios in planetary systems have In this paper, we extend such studies, considering systems with or- operated. This is because the probability of forming resonant con- bital parameters similar to those of HD 155358 (Robertson et al. 2012a), 24 Sextantis (Johnson et al. 2011), HD 60532 (Laskar & Correia 2009) as well as HD 6805 (Trifonov et al. 2014) each E-mail: [email protected] of which have the inner planet with semi-major axis in the range

C 2016 The Authors Published by Oxford University Press on behalf of the Royal Astronomical Society A pair of giant planets in MMR 4407

Table 1. Properties of the HD 155358, 24 Sextantis, HD 6805 and HD 2 BASIC EQUATIONS 60532 systems. The first three either are or possibly in 2:1 resonance while θ ϕ the fourth is in a 3:1 resonance. The first column identifies the system, and We adopt a spherical coordinate system (r, , ) with associated the second column gives the mass of the central star in solar . The unit vectors (rˆ, θˆ, ϕˆ ) and origin at the centre of mass of the central third and fourth columns give the masses of the planets in masses star. and the fifth and sixth columns give their semi-major axes in au. The basic equations governing the disc express the conserva- tion of mass and momentum under the gravitational potential due System M∗ M1 M2 a1 a2 to the central star and any planets and incorporate a kinematic viscosity ν, HD 155358 0.92 0.85 ± 0.05 0.82 ± 0.07 0.64 1.02 24 Sextantis 1.54 1.99 ± 0.4 0.86 ± 0.4 1.33 2.08 ∂ρ =−∇· ρv , HD 6805 1.7 2.5 ± 0.2 3.3 ± 0.2 1.27 1.93 ∂t ( ) (1) HD 60532 1.44 3.15 7.46 0.77 1.58 Dv ρ =−ρ∇ −∇P +∇· T . Dt (2) Here the convective derivative is defined through 0.5–1.4 au. We perform 2D and 3D numerical simulations of pairs D ∂ ≡ + v ·∇, Downloaded from of giant planets interacting with a protoplanetary disc that attain a Dt ∂t (3) mean motion commensurability for up to 2.5 × 104 orbital peri- ods of the inner planet. We investigate whether such systems could ρ is the density, v is the velocity, P is the pressure, T is the viscous have originated at larger radii beyond the ice line and then migrated stress tensor (see e.g. Mihalas & Weibel Mihalas 1984), and  is the inwards, the commensurability possibly being formed in the same gravitational potential which has contributions from the central star

neighbourhood. and any planets present. Disc self-gravity is neglected. The pressure http://mnras.oxfordjournals.org/ In order to study the role of the nature of the underlying disc is related to the gas density and the isothermal speed of sound cs P = ρc2. model, we consider models with an inner magnetorotational insta- through s bility (MRI) active region producing a significant effective viscosity and an outer inactive region for which a significant effective vis- 3 NUMERICAL SIMULATIONS cosity may occur only in the upper layers of the disc (Gammie 1996) as well as models with a uniform α-viscosity prescription Simulations were performed using the finite volume fluid dynamics throughout. We also consider models with different surface density code PLUTO (Mignone et al. 2007) which has been used success- scaling corresponding to varying the total disc mass or equivalently fully to simulate protoplanetary discs interacting with planets (e.g. the steady-state accretion rate. In this way, the disc–planet interac- Mignone et al. 2012; Uribe, Klahr & Henning 2013). The planet at University of Cambridge on August 10, 2016 tion at different stages of the life of the protoplanetary disc can be positions are advanced using a fourth-order Runge–Kutta method studied with lower mass discs corresponding to later stages (e.g. which however assumes that the forces due to the disc do not change Calvet et al. 2004). as the planet locations are advanced through a time step, making We find that when low-mass disc models are considered, systems the method one of lower order (see e.g. Nelson et al. 2000 for with planets in the Jovian mass range maintain a 2:1 commensu- comparison). For some runs, we employed the FARGO algorithm rability while undergoing inward type II migration. This is found of Masset (2000) as this allows the numerical calculations to run to be at a rate such that formation at a few au from the central significantly faster. We note that for our simulations the options star and migration to their current locations on a time-scale com- were chosen such that algorithm was applied using the residual parable to the expected protoplanetary disc lifetime is possible in azimuthal velocity with respect to the initial azimuthal velocity, principle. the latter not being updated. A sample of runs were checked care- We find that there is a relative insensitivity of results to the fully to confirm numerical stability and that consistent results were disc model employed and find good agreement between 2D and 3D obtained (see also Mignone et al. 2012 for a comparison of this simulations. Planets containing larger masses such as the HD 60532 type). system which is observed to be in 3:1 resonance (Laskar & Correia For the calculations reported here, we adopt a locally isother- −1/2 2009) are found to attain this resonance in low-mass low-viscosity mal equation of state for which cs ∝ r . The constant of pro- discs. portionality is chosen so as to give a constant aspect ratio h ≡ We find that for systems with planets in the Jovian mass range, H/r = cs/(r K) = 0.05. Here H is the disc semi-thickness and K increasing the disc mass results in the formation of an unstable is the local Keplerian circular orbit angular velocity. We adopt a 5:3 resonance. This instability results in the rapid destruction of system of dimensionless units such that masses are expressed in the commensurability implying that the occurrence of such systems units of the central M∗, radii are expressed in units should be less common. of the initial orbital radius of the innermost planet and times are The plan of this paper is as follows. We give the basic equations expressed in units of the of a circular orbit at that and coordinate system used in Section 2. In Section 3, we outline radius. the numerical methods and computational domains adopted going For three-dimensional simulations, the radial computational do- on to describe aspects of the physical set-up and disc models used main in most cases was given in dimensionless units by r ∈ [0.15, in Sections 3.1 and 3.1.1. We then indicate how results might be 3.75]. The grid outer boundary is treated as a rigid boundary and scaled to different radii and summarize important aspects of type taken sufficiently far from the planets so that the density perturba- II migration in Sections 3.2 and 3.3. We go on to describe our tions they create in the disc are damped before they reach it, while numerical results in Section 4, beginning with a comparison with the grid inner boundary only allows inflow, so that the disc material previous results for two migrating planets presented in section 4.1. can be accreted on to the central star. The θ domain was taken to be Finally, we discuss our conclusions in Section 5. [π/2 − 3H/r,π/2] ≡ [θmin, π/2], with symmetry being assumed

MNRAS 461, 4406–4418 (2016) 4408 Q. Andre´ and J. C. B. Papaloizou with respect to the plane θ = π/2, this being treated as a rigid boundary. The ϕ domain was taken to be [0, 2π]. The standard grid resolution for most simulations was taken to be (Nr, Nθ , Nϕ ) = (162, 18, 314). The radial grid spacing was non-uniform and chosen so that the grid spacing r was equal to 0.02 r. This geometric spacing is the most natural one, since the disc semi-thickness scales as r. The azimuthal grid spacing was uniform Figure 1. Illustration of our disc model, which contains an inner MRI active and such that ϕ = r/r = 0.02. We remark that disc–planet in- zone (AZ) together with an outer disc with reduced activity. This may have teractions adopting similar resolutions to those adopted here have a dead zone (DZ) together with an active surface layer. The transition radius been carried out for lower mass planets by Kley, Bitsch & Klahr between those two regions is located at 0.6 au. For standard runs, α does not θ α (2009). The interval θ was chosen such that there were six grid vary with or height in the disc. The value of then decreases sharply from α = −3 α = −4 cells per scaleheight. Planet gravitational potentials were softened AZ 10 to DZ 10 as the transition radius is passed through. For the layered model, α is only non-zero for [π/2 − 3H/r,π/2] ≡ [θ , π/2] by adopting a gravitational softening length taken to be 0.5 Hill min in the outer disc. radii, being slightly less than two radial grid cells in extent, and the smoothing filter of Crida et al. (2009a) was employed when cal- of the range of accretion rates observed for protoplanetary discs culating torques acting on planets. We also performed convergence

(Calvetetal.2004) and might be expected to occur during their late Downloaded from checks using simulations for which Nr was increased to 229, Nθ stages which are the focus of attention here. However, we have also increased to 25 and Nϕ was√ increased to 444, with the softening considered disc models with surface density scaled such that they . length reduced by a factor of 2 are up to 10 times more massive and accordingly with a steady-state For two-dimensional simulations, in most cases the radial and az- accretion rate that is also up to 10 times higher. imuthal grids and domains are the same as in the three-dimensional To allow for an inner MRI active region together with an outer

θ π/ . ρ http://mnras.oxfordjournals.org/ case but now is fixed to be 2 In this case, the mass density region with much less activity, we specify α to be a function of r. is replaced by the surface density and the pressure is replaced by Thus ν = αcsH,where a vertically integrated pressure. The gravitational softening length α − α was taken to be 0.6H and the smoothing filter of Crida et al. (2009a) AZ  DZ  α = αDZ + r . (4) was employed when calculating torques acting on planets for this + − − 1 exp 25 1 . case also. The convergence of two-dimensional simulations was 0 6 checked by performing them at twice the resolution with the soft- Here αAZ corresponds to the inner region and αDZ corresponds to the −3 −4 ening procedure remaining fixed. outer zone. Our standard model has αAZ = 10 and αDZ = 10 . As we consider either low-viscosity discs or discs undergoing The functional form of α gives a sharp transition around r = 0.6. steady-state accretion at very low accretion rates, we neglect ac- The surface density is chosen to provide the prescribed accretion − / at University of Cambridge on August 10, 2016 cretion on to the planet. Many of our runs are carried out with rate through the relation M˙ = 3πν .Thisgives ∝ r 1 2 when α planets migrating in discs with local kinematic viscosity ν<10−6 is constant. in dimensionless units. Accretion on to the planet has been found to be a small effect in this case (Bryden et al. 1999; Kley 1999). In addition, the time required for the accretion rate through the disc to 3.1.1 Layered model double the planet’s mass is significantly longer than the migration In order to consider the possibility that a significant effective disc time when larger viscosities are considered (see below). viscosity arises only in the upper layers of the outer disc (e.g. Gammie 1996), we have performed simulations for a layered outer 3.1 Disc model and viscosity prescription disc model. In this model, α was taken to be non-zero only in the upper half of the θ domain, namely [θ , (θ + π/2)/2] (see For global simulations that need to be run for long times such as min min Fig. 1). The value of α in the upper domain was chosen to be such those we perform, constraints on numerical resolution are such that that the integrated stress in the meridional direction was the same small-scale turbulence associated with angular momentum transport as for an outer disc model with α independent of θ (see Pierens & has to be modelled through an effective viscosity prescription. To Nelson 2010). In practice, the value of α in the upper layers of the do this, we adopt an α-viscosity prescription (Shakura & Sunyaev layered model was thus found to be 7.5 times larger than the value 1973). For this, the kinematic viscosity ν = αc H. The value of α s for the corresponding non-layered model. to be adopted depends on the expected level of turbulence and this For three-dimensional models, the disc equilibrium pressure is in turn depends on the operation of the MRI (Balbus & Hawley given by 1991). It is expected that there will be an inner MRI active zone   together with an outer dead zone which may have active surface GM∗ P = P (R)exp ln(sin θ) . (5) 0 c2r layers (Gammie 1996). Small-scale hydrodynamic instabilities such s as the vertical shear instability may also contribute in the absence Here R = r sin θ and the function P can be chosen to match a of the MRI (Nelson, Gressel & Umurhan 2013). The location of the 0 prescribed surface density or alternatively mid-plane pressure or interface between these regions depends on details of the transition density. from magnetohydrodynamic stability to instability and is subject to some degree of uncertainty. Latter & Balbus (2012) estimate that it typically occurs at around 0.6 au. This is illustrated in Fig. 1. 3.1.2 Initial gaps We specify the standard disc model to be such that when the unit of length used to scale our dimensionless units is 1 au and the unit In the simulations presented here, the density or surface density of mass is a solar mass, it corresponds to a steady-state model with was modified such that any planets were initially placed within accretion rate M˙ = 6.0 × 10−10 M yr−1. This is near the bottom gaps. The initial orbital evolution would differ if the planets were

MNRAS 461, 4406–4418 (2016) A pair of giant planets in MMR 4409 initially embedded. However, because our goal is to measure steady- 4 NUMERICAL RESULTS state migration rates, this is not a problem. Thus, the simulations are assumed to commence after any gaps have been formed. In this 4.1 Comparison with previous results way, we avoid having to simulate an uncertain initial phase during Because our simulations of planet–disc interaction with the PLUTO which gaps are formed. code were implemented from scratch, we begin by establishing that The procedure we adopted was to reduce the density or surface some of the main results obtained in previous studies are recovered density by a constant factor in the interval [r /1.15, 1.15r ]where p p with our code. In particular, we focus on the much studied GJ 876 r is the initial orbital radius of the planet, being assumed to be in p system (see e.g. Snellgrove, Papaloizou & Nelson 2001; Kley et al. a circular orbit. This profile was then joined to the original through 2005;Crida,Sandor´ & Kley 2008), as well as a case involving the linear connections in the intervals [0.9r /1.15, r /1.15] and [1.15r , p p p action of the so-called Masset–Snellgrove mechanism invoked to 1.265r ]. The gap reduction factor was taken to be a factor of 10 p reverse type II migration (Masset & Snellgrove 2001) that has been in most cases, being a factor of 100 for the cases with planets with considered by Crida, Masset & Morbidelli (2009b). final masses exceeding 2 Jupiter masses. In addition, the planets were held in fixed circular orbits for 200 orbits before being released. Their masses were built up to their 4.1.1 The case of GJ 876 final values over the first 20 orbits using the procedure given by de Val-Borro et al. (2007). Our first comparison run was initiated with the mass ratio equal to Downloaded from 6 × 10−3 for the inner planet, and 1.8 × 10−3 for the outer planet. These parameters correspond to those of GJ 876. The simulation employed the physical set-up described in Snellgrove et al. (2001, 3.2 Scaling to arbitrary radii see their section 3.1). A constant aspect ratio h = 0.07 and a constant α-viscosity prescription with α = 2 × 10−3 are adopted. The two The results obtained with the unit of radius chosen to be the initial http://mnras.oxfordjournals.org/ planets are initially in circular orbits, and start their evolution with orbital radius of the inner planet can be scaled to apply to arbi- semi-major axis a = 1.0 for the outer planet and a = 0.6 for trary radii. This is done by noting that results are invariant if the 1 2 the inner planet. Note that for this comparison, we adopt their length-scale is multiplied by λ, the time-scale is multiplied by λ3/2 nomenclature and system of dimensionless units. Hence, the outer and the mass scale left unaltered. To be consistent with this, the planet is initially located outside the exact 2:1 commensurability surface density should be reduced by a factor λ2. If regions of the (a ∼ 0.95). disc are connected in this way, the situation does not correspond 1 A putative uniform initial disc surface density corresponding to a steady-state disc. However, this is not unreasonable if it is 0 to what would give a disc mass of 2M within the initial orbit of applied at radii where the age or evolution time of the system is J the outer planet was taken. It is also assumed that both planets are less than the local viscous time-scale. For our standard disc model, at University of Cambridge on August 10, 2016 located inside a tidally truncated cavity located at r < 1.3, with low this would be the case for length-scale exceeding 2 au and age surface density equal to 0.01 . In the region 1.3 < r < 1.5, the  2 × 106 yr. 0 surface density is prescribed such that ln linearly joins to ln . We note that when applied, the scaling procedure shifts the transi- 0 In addition in our run, the smoothing filter of Crida et al. (2009a) tion radius between the active and inactive regions while its location was used when calculating torques acting on planets. should in principle remain constant. However, this is not a problem The results are displayed in Fig. 2 which can be compared with as we find an insensitivity of our results to the location of the tran- fig. 1 of Snellgrove et al. (2001). The two planets first undergo a sition radius as long as the planets migrate in the outer disc (see phase of convergent migration. At time t ∼ 400 orbits, the period below). ratio between the two planets locks around the value 2, and the resonant angle 2λ2 − λ1 −  1 starts to librate around 0. Here λ2, λ1,  2 and  1 are the longitudes of the inner and outer planet and the 3.3 Aspects of type II migration longitudes of pericentre for the inner and outer planets, respectively. A 2:1 mean motion resonance is subsequently maintained. The The planets in the simulations presented here are massive enough behaviour we obtain is very similar to that found by Snellgrove to make deep gaps in the disc surface density profile. Accordingly, et al. (2001) until a run time 1500 orbits is reached. After this, the they undergo type II migration (e.g. Lin & Papaloizou 1986,Lin evolution of the semi-major axes and eccentricities stall in their run & Papaloizou 1993; Baruteau et al. 2014). The rate of migration while they continue to respectively decrease and increase slightly in is governed by the viscous time-scale and the disc mass within a ours. We believe that the stalling is artificial and due to the approach radial scale comparable to its orbital radius, r. Baruteau et al. (2014) of the inner planet to the inner boundary, located at a radius equal to τ −1, estimate the migration rate of a planet of mass Mi as m where 0.4 in their case. In order to avoid this, we chose an inner boundary   radius rin = 0.2. This allowed us to continue the evolution of the Mi τ = τν max 1, , (6) two planets for up to 3000 orbits without this type of influence m π r2 4 from the inner boundary. In the context of the above discussion, we with τ ν being a viscous time-scale. Thus, when the planet mass remark that Kley et al. (2005) performed simulations with the inner becomes large, the evolution rate slows on account of the inertia of planet totally interior to the calculation domain. They found that its the planet. Note that Ivanov, Papaloizou & Polnarev (1999) obtain eccentricity continued to increase as we did (see their fig. 10). a faster rate than implied by equation (6), finding that the second term in the parentheses appears taken to a fractional power. The rate 4.1.2 An example illustrating the Masset–Snellgrove mechanism is faster because disc material tends to pile up near the outer gap edge increasing the angular momentum flux that the planet needs Masset & Snellgrove (2001) found that the migration of two planets to provide in order to maintain it. locked in a mean motion resonance can proceed outwards. For this

MNRAS 461, 4406–4418 (2016) 4410 Q. Andre´ and J. C. B. Papaloizou Downloaded from http://mnras.oxfordjournals.org/ at University of Cambridge on August 10, 2016 Figure 2. Results from the GJ 876 comparison run. The uppermost panel Figure 3. Results from the run illustrating the Masset–Snellgrove mecha- shows the evolution of the semi-major axes of the two planets, the middle nism. The panels correspond to those of Fig. 2. panel shows the eccentricities (the upper curve corresponding to the inner planet), the left-hand bottom panel shows the period ratio for the two planets to migrate inwards. The outer planet moves rapidly in a type III λ − λ −  and the right-hand bottom panel shows the resonant angle 2 2 1 1. migration regime with the inner one moving much more slowly The unit of time, t, in this and subsequent figures is the orbital period at a corresponding to type II migration. This convergent migration of the value of the dimensionless radius equal to unity. two planets causes passage through the 2:1 mean motion resonance. The resonant angle 2λ2 − λ1 −  1 starts librating around zero from to occur, they must open overlapping gaps and the outer planet be of time t ∼ 1000. Subsequently, the convergent migration stops and significantly lower mass than the inner one. Our second comparison the two planets migrate smoothly outwards together. run was chosen such that this mechanism is expected to operate. It The qualitative behaviour described above is very similar to that was initiated with mass ratios for the inner and outer planets respec- obtained by Crida et al. (2009b). However, they obtain an acceler- tively equal to 3 × 10−3 and 10−3. These parameters correspond to ation at early times followed by a significant slow-down later on, those adopted by Crida et al. (2009b). The simulation adopted their whereas in our case the outward migration rate is more uniform. As physical set-up apart from the treatment of the boundaries (see their indicated above, this difference is not unexpected on account of the section 3.2). They employ an additional matched one-dimensional role of the inner boundary. Thus, we find that just after the planets simulation of a putative enveloping disc, whereas we adopt our start moving outwards, the inner planet initially migrates outwards standard conditions described above. As the inner boundary radius at half the rate obtained by Crida et al. (2009b). However, after 5 is located at a radius equal to 45 per cent of the initial inner planet 5000 orbits at r = a0 (corresponding to a time t ∼ 1.12 × 10 yr in orbital radius and the interior disc plays a major role in driving the Crida et al. 2009b), when the inner boundary is expected to be less outward migration, we expect differences in results at early times. important, the mean migration rates are approximately the same. The outer boundary may also become significant at late times. The form of the aspect ratio adopted is h = 0.045 × (r/a )1/4 (a being 0 0 4.2 The standard run the initial inner planet orbital radius), the initial surface density is −3/2 −3 (r) = 0 × (r/a0) with 0 = 1.5 × 10 and an α-viscosity We now consider the runs that are the focus of this paper which, prescription with α = 0.01 is adopted. The two planets are held in unlike in Section 4.1.2, involve for the most part inward conver- circular orbits for the first 100 orbits of the inner planet, and start gent migration of the planets. For these cases as well as that of GJ their evolution with semi-major axis a1 = 1 = a0 for the inner planet 876, an overview can be obtained by considering a simple N-body and a2 = 2 for the outer planet. model in which the planets move as particles under their gravita- The results are displayed in Fig. 3 which can be compared with tional interaction and the influence of additional forces presumed to fig. 1 of Crida et al. (2009b). After the release, both planets start arise from interacting with the disc (see e.g. Snellgrove et al. 2001;

MNRAS 461, 4406–4418 (2016) A pair of giant planets in MMR 4411 Downloaded from http://mnras.oxfordjournals.org/

Figure 4. Results for the 2D standard run. The panels correspond to those at University of Cambridge on August 10, 2016 of Fig. 2.

Lee & Peale 2002; Nelson & Papaloizou 2002). These result in orbital circularization and migration. It is found that unless the convergent migration is very rapid, the planets attain a commen- surability and then migrate together maintaining it. As they do so, their eccentricities increase until either their migration halts or their rate of growth can be balanced by a damping process. Figure 5. The upper panel shows surface density contours in logarithmic For the slowest migration rates, a 2:1 resonance is attained for scale for the disc after 280 orbits of the simulation shown in Fig. 4.The Jovian mass planets while for larger masses a 3:1 resonance can be lower panel shows the azimuthally averaged surface density in logarithmic attained. As the rate of convergence is increased, closer commensu- scale after 3750 orbits (solid curve) and the initial surface density (dashed rabilities are attained. Our results are fully in line with these general curve). The planets occupy the deep common gap. expectations. The rate of convergent migration and hence the close- ness of the commensurability is determined by the rate of angular momentum transport in the disc which for a fixed α distribution and outer planet, respectively. An ultimate libration amplitude of a increases with the mass in the disc. few per cent is indicated. The lowermost right-hand panel of Fig. 4 The 2D standard run was initiated with the mass ratio for both shows the resonance angle, 2λ2 − λ1 −  1, appropriate for the 2:1 planets equal to 10−3. Noting that they are somewhat uncertain, resonance. This resonance angle ultimately librates around zero. these parameters may approximately correspond to those for HD The behaviour of the second resonance angle, 2λ2 − λ1 −  2,is 155358 and 24 Sextantis (see Table 1). The simulation was initiated very similar. with the planets occupying a common gap in the standard disc as At late times, the migration time −r/r˙ ∼ 1.5 × 105 inner planet described above. We assume that they start their evolution with initial orbital periods. For comparison, the viscous time 2r2/(3ν) 5 semi-major axis a1 = 1 for the innermost planet and a2 = 1.7 for for the outer disc is 4.2 × 10 orbits at r = 1 in dimensionless units. the outermost planet in dimensionless units. The results are plotted Note that the two resonantly coupled planets migrate at a similar in Fig. 4. The evolution of the semi-major axes and eccentricities rateasasingleplanet(seebelow). are shown for a time interval of 2.5 × 104 time units. The system Additional results from the 2D standard run are illustrated in enters 2:1 resonance after a few hundred orbits. Note that in this Fig. 5. The upper panel shows surface density contours for the disc and other cases, the inner planet migrates outwards at early times after 280 orbits and the lower panel shows the azimuthally averaged on account of the influence of the inner disc. The lowermost left- surface density gap after 3750 orbits. Note the non-axisymmetric hand panel shows the evolution of the orbital period ratio n1/n2 of vortex-like structures in the low surface density ring between the the two planets, n1 and n2 denoting the mean motions of the inner planets (see e.g. de Val-Borro et al. 2007). The planets occupy a

MNRAS 461, 4406–4418 (2016) 4412 Q. Andre´ and J. C. B. Papaloizou

Figure 6. Evolution of the semi-major axis of a single planet with mass ratio 10−3 migrating in a standard disc plotted together with the evolution of the semi-major axis for the innermost planet in the standard run. The evolution is shown as a function of time expressed in units of the orbital period at r = 1 in dimensionless units. Downloaded from deep common gap that is characteristic of the simulations reported here. In Fig. 6, we illustrate the migration of a single planet in a standard disc. For comparison, the evolution is plotted together with that for the inner planet in the standard case. At the beginning, the evolution in the standard case is outwards. This is because the planet starts http://mnras.oxfordjournals.org/ in a much wider gap and is pushed outwards by the inner disc in that case. However, once the system attains resonance, the planet is pushed inwards by the inwardly migrating outer planet. For the isolated planet, the inward migration is driven by the outer disc directly. In both cases, we expect a characteristic migration rate corresponding to type II migration and indeed the rates are found to be ultimately comparable. The migration speed of the single migrating planet attains a steady value between 5000 and 8000 Figure 7. Results of a simulation with the same conditions as for the 2D orbits. The mean migration time-scale over this period is estimated standard run except that only an active model disc was used. That is the at University of Cambridge on August 10, 2016 as −r/r˙ ∼6.9 × 104 orbits at a dimensionless radius of unity. transition radius of r = 0.6 for the standard run can be regarded as being This is characteristic of type II migration, an aspect that is discussed moved to a very large value. The panels correspond to those of Fig. 4.Note further in Section 5. that the migration is significantly slower than in the standard case.

the same time. Note that the eccentricities at times corresponding to 4.2.1 An entirely active disc the same amount of relative joint migration are smaller in this case Fig. 7 shows results for a simulation with the same conditions, than those for the corresponding standard case, which is indicative including the initial set-up of the pair of planets, as for the 2D of a larger damping rate. standard run, except that only an active model disc was used. That is the transition radius of r = 0.6 for the standard run was effec- tively moved to very large radii. The quantities plotted in the various 4.2.3 Three-dimensional simulations panels of Fig. 7 correspond to those plotted in the corresponding We now consider 3D simulations. The disc models were set up panels of Fig. 4 and the results are qualitatively very similar. The as described in Sections 3.1 and 3.1.1 with the planets introduced −r/r ∼ . × 5 estimated late-time migration time ˙ 2 4 10 orbits is sig- together with initial gaps as described in Section 3.1.2. We recall nificantly shorter than in both the standard two-planet case and the that for the standard 3D model α does not vary with θ. single-planet case where the planet migrates in the inactive region. A comparison of the evolution of the semi-major axes for the This is because even though the viscosity is 10 times larger in the standard 2D two planet with the corresponding results from the active disc, the surface density is 10 times smaller resulting in a standard 3D run is given in the upper panel of Fig. 9. The evolution slower migration rate on account of the reduced disc mass in the of the eccentricities is shown in the lower panel. These are found neighbourhood of the planets. to be in very good agreement with each other and therefore also in accord with the simplified N-body approach mentioned above. This suggests that the evolution can be determined by considering the 2D 4.2.2 A disc without an inner active region response of the vertically averaged disc. The fact that the response In Fig. 8, we illustrate a 2D two-planet run with the same parameters is for the most part two-dimensional is indicated by the behaviour as the standard run except that the inner active disc was removed or of the disc state variables. We show density contours in logarithmic equivalently the transition radius was moved to very small values. scale for three indicated values of θ at a typical late time of 1100 The quantities shown in the panels of Fig. 8 are the same as in the orbits after the start of the simulation in Fig. 10. These values cor- corresponding panels of Fig. 4. The migration time-scale at later respond to the mid-plane and approximately to heights H and 2H times is estimated to be −r/r˙ ∼ 105 orbits at a dimensionless radius above it. It will be seen that apart from in the neighbourhoods of the of unity. This is almost exactly the same as for the standard case at planets, the density distributions are approximately the same at the

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Figure 9. Comparison between the 2D and 3D standard runs. A comparison

of the evolution of the semi-major axes is given in the upper panel. The lower http://mnras.oxfordjournals.org/ panel shows the evolution of the eccentricities with the upper pair of curves corresponding to the inner planet.

of gravity of the planet are apparent. This is an effect that can only be represented in 3D. However, this is not found to cause significant departures from the 2D results for the orbital evolution, presumably because of the small amount of material in the gap regions near to Figure 8. As for the 2D standard run shown in Fig. 4 except that in this the planets. We remark that a similar behaviour of the state vari-

case the disc model was such that the inner active disc was removed or ables to that described here was found by Pierens & Nelson (2010) at University of Cambridge on August 10, 2016 equivalently the transition radius can be regarded as being moved to a very in their 3D simulations with a single planet. small value. The panels correspond to those of Fig. 4. different heights apart from a constant scaling factor. In the neigh- 4.2.4 Layered model bourhoods of the planets, there are the local mass concentrations usually seen in 2D simulations (see the left-hand panel of Fig. 10). A comparison of the results obtained for the layered model described In the upper regions of the disc, these are absent and there is instead in Section 3.1.1 with those obtained from the standard 2D model is material depletion on account of vertical flows towards the planets. illustrated in Fig. 12. The layered disc model had the same integrated To illustrate this aspect, Fig. 11 shows the characteristic form of stress in the meridional direction as in the standard case. Apart the stream lines in a meridional section at the azimuth of the inner from the differing disc model, the conditions are the same as for planet. These are shown after a time corresponding to 1410 orbits the standard 3D run. It will be seen that the evolution of the semi- at dimensionless radius equal to unity. Significant vertical motions major axes for the two runs is almost identical. We remark that the associated with material moving from the upper regions of the disc migration rates for both planets are slightly faster in the standard towards the mid-plane slightly interior to the location of the centre case as compared to the simulation with the layered model. This is

Figure 10. Contours of log ρ in dimensionless units in the (r, ϕ) plane are shown for three indicated values of θ for the 3D standard run after 1100 orbits. From left to right, these values correspond to the mid-plane and approximately to heights H and 2H above the mid-plane.

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Figure 11. Stream lines in a meridional section at the azimuth of the loca- tion of the inner planet for the 3D standard run after 1410 orbits. The colour scale indicates log ρ in dimensionless units and the dashed circle indicates the location of the surface of the Hill sphere. Downloaded from http://mnras.oxfordjournals.org/

Figure 13. Results for a 2D two planet run with the same parameters as for the standard run, except that the initial disc model was modified such that at University of Cambridge on August 10, 2016 the surface density was increased everywhere by a factor of 5. The panels correspond to those of Fig. 4, except for the lowermost right-hand panel which shows the resonant angle 5λ2 − 3λ1 − 2 1, appropriate for the 5:3 resonance. Note that the middle panel shows the evolution of eccentricities, with that of the inner planet initially mostly smaller but with the curves for the two planets later overlapping. This run eventually goes unstable after approximately 5000 inner planet orbits. Figure 12. Results for the 3D layered model compared to those of the 2D standard run. The panels correspond to those of Fig. 9. Note that the lower panel shows the evolution of the eccentricities with the uppermost pair of ties, the left lowermost panel shows the period ratio and the right curves corresponding to the inner planet. lowermost panel shows a resonant angle. The outer planet initially migrates rapidly in a type III migration regime while the semi-major axis of the inner planet hardly changes, in line with the results of Pierens & Nelson (2010) for the case of a indicating an approximate balance between inward and outward single Jupiter mass planet. migration torques. As a consequence, the planets rapidly enter a 2:1 However, the eccentricities are significantly larger at correspond- resonance which quickly develops an instability, as indicated by the ing times for the layered model. This indicates that the eccentricity strong fluctuations in eccentricity and period ratio. The instability damping rate is lower in this case and accordingly it depends on causes the planets to leave the resonance at around t = 1000 and the detailed properties of the disc model. For the layered model, we resume convergent migration. During this phase, the inner planet recall that the disc is inviscid near the mid-plane. migrates slowly outwards on account of the action of the inner disc and a deeper common gap. The planets enter a 5:3 resonance at t ∼ 2000. This persists until time t ∼ 5000. Accordingly, the resonant 4.3 Increased disc surface density and the formation angle shown is 5λ − 3λ − 2 . This librates about π once the of a 5:3 resonance 2 1 1 resonance forms. The angle 5λ2 − 3λ1 − 2 2 shows very similar In order to investigate the effect of increasing the disc surface den- behaviour. Although the estimated inward migration time −r/r˙ ∼ sity, we performed 2D two-planet runs with the same parameters 50 000 inner planet initial orbital periods, while the system is in as the standard run, except that the disc model was modified such 5:3 resonance this run eventually goes unstable after approximately that the surface density was increased by a factor of 5. From the 5000 orbits culminating in the two planets undergoing a scattering discussion of the disc models in Section 3.1, this corresponds to (see also Lee, Thommes & Rasio 2009, for an approach based on increasing the steady-state accretion rate by the same factor. N-body methods). Thus, observing a system in 5:3 resonance is The results for this case are illustrated in Fig. 13. The uppermost unlikely. We remark that although the planet mass ratios differed panel shows the semi-major axes, the middle panel the eccentrici- and the transition was from a 2:1 resonance to a 3:2 resonance,

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Figure 14. A comparison of two 2D runs which were identical to the Figure 15. Results for a 2D run which was identical to the standard case standard case except that the values of α in the dead zone were taken to be except that the value of α in the active zone was taken to be 10−2. The panels 2 × 10−4 and 5 × 10−5, respectively. The panels correspond to those in correspond to those in Fig. 4. at University of Cambridge on August 10, 2016 Fig. 4. The period ratio and resonant angle plots overlap for these cases. outer inactive region was increased and decreased by a factor of 2 as qualitatively similar behaviour to that described above, up until compared to the standard run, while the surface density was scaled the instability at late stages, was found in simulations of Rein, such as to maintain the same steady-state accretion rate as for the Papaloizou & Kley (2010, see their fig. 6). standard run. The inner active disc region was the same as in the In order to investigate further the effect of increasing disc mass, standard case. The quantities plotted in the different panels are as we performed an additional simulation for which the disc mass was in Fig. 4. We see that the migration rate decreases with decreasing increased by a factor of 10. In this case, the surface density is large viscosity as expected but somewhat less steeply than linearly. This enough that the Toomre criterion for gravitational instability is close is because the migration rate has a dependence on the disc surface to being marginal at the outer boundary, that being potentially the density profile which takes a long time to evolve, especially in the most unstable location. As self-gravity has been neglected, this disc case with the smallest viscosity. model provides the maximum strength of the disc planet interaction In Fig. 15, we present a 2D run for which the initial conditions that we can consider. We investigate it in order to check that it is were as in the standard case except that the value of α in the inner the model for which the closest commensurability is attained as active region was increased from 10−3 to 10−2. The quantities plot- indicated by the discussion at the beginning of Section 4.2. ted in the different panels are as in Fig. 4. The results on this case This case behaves similarly to the previous one except that the are very similar to those obtained for the standard run. We remark initial migration phase is more rapid and such that the system passes that the planets are migrating in the inactive part of the disc. Ac- straight through the 2:1 resonance before going into an unstable cordingly, this result indicates that the inner active disc plays only 5:3 resonance. After 2000 orbits, it then undergoes a transition to a minor role in these circumstances. a stable 3:2 resonance where it remains until 10 000 orbits after the start of simulation. While the system in the 3:2 resonance, the mean migration rate is approximately 50 per cent faster than for 4.5 A case with a more massive outer planet the previous case. As expected, the closest commensurability was We have also considered the effect of increasing the mass of the outer attained in this case. planet. In Fig. 16, we give the results for a simulation for which the mass ratio of the outer planet was increased by a factor of 2 and its starting distance was increased from 1.7 to 2 in dimensionless units 4.4 The effect of changing the magnitude of the viscosity otherwise initial conditions are as in the standard model. However, In order to examine the effect of changing the magnitude of the vis- the surface density was everywhere taken to be 78 per cent of the cosity, a comparison of the semi-major axis evolution for different standard value. This was done so as to make the disc mass in the 2D runs is given in Fig. 14. For these runs, the value of α in the neighbourhood of the outer planet the same as in the standard case.

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Figure 16. As for the standard case shown in Fig. 4 except that the surface density in the initial disc model is reduced through multiplication by a factor Figure 17. Results for a 2D two-planet run where the outcome is migration

= at University of Cambridge on August 10, 2016 0.78. In addition, the mass of the outer planet which starts further out at r 2 in a 3:1 resonance. The parameters were chosen to correspond to the HD is increased by a factor of 2. 60532 system (see text), with the outer planet starting at r = 2.25. The panels correspond to those in Fig. 4 except that the resonant angle plotted is λ − λ −  In addition, the grid outer boundary was shifted from 3.75 to 4.4 such 3 2 1 2 1. that it was kept distant enough from the outermost planet. Bearing in mind the observational uncertainties, the mass ratio being larger of their eccentricities. That in turn can be traced to a tendency of the for the outer planet in this system allows it to potentially resemble inner planet to increase its rate of migration inwards at late times the HD 6805 system. This run attains a 2:1 resonance and migrates unlike in most other 2D cases where an inner planet is pushed by inwards with the two resonant angles ultimately librating around an outer planet. Note that in this case the difference is that the inner zero and such that −r/r˙ ∼ 105 inner planet initial orbital periods. planet is more massive and so more effective at clearing away the This characteristic time is close to that found for the standard run inner disc which opposes its inward migration. and so a very similar discussion will follow (see Section 5). We remark that this system has been considered by Sandor´ & Kley (2010). They considered a disc with the same aspect ratio and approximately the same mass interior to the inner planet as consid- 4.6 The formation of a 3:1 resonance ered here. However, the viscosity parameter α = 0.01 throughout We have also studied a case with planet masses chosen to correspond corresponding to a much larger viscosity than we consider for our to the HD 60532 system (see Table 1) where the planets have been standard model disc. They found that the planets attained a 3:1 res- −r/r ∼ found to be in a 3:1 resonance (see Laskar & Correia 2009). onant configuration with ˙ 4500 orbits. In our case, this time The results of the simulations are shown in Fig. 17. The quan- is extended to 60 000 initial inner planet orbits. This is discussed tities plotted in the different panels correspond to those in Fig. 4 further below. except that the right lowermost panel shows the resonant angle 3λ − λ − 2 . This is found to exhibit large-amplitude libra- 2 1 1 5 SUMMARY AND DISCUSSION tions about π as does the angle 3λ2 − λ1 − 2 2. The behaviour of the angles that we find here is similar to that presented by Laskar & In this paper, we have performed 2D and 3D simulations of pairs Correia (2009). Note that the two planets speed up their joint inward of giant planets that have attained a mean motion resonance in a migration slightly between t = 5000 and 7000 while their mean ec- protoplanetary disc. We considered disc models both with an in- centricities decrease slightly. As the rate of growth of eccentricity in- ner active region and an outer inactive region with lower effective creases with the rate of convergent migration as the planets become viscosity as well as disc models incorporating only one of these closer to resonance (e.g. Papaloizou 2003; Baruteau & Papaloizou regions. Different magnitudes for the viscosity in both regions were 2013), this is an indication that the planets are tending to converge considered. This was found to have only minor effects on the results more slowly rather than there being an increase in the damping rate as long as the surface density was scaled such as to maintain the

MNRAS 461, 4406–4418 (2016) A pair of giant planets in MMR 4417 same steady-state accretion rate – or equivalently, outward directed potential importance of a residual inner gaseous disc for damping the angular momentum flux, and this scaling did not result in the sur- eccentricity of the inner planet and so preventing the eccentricities face density becoming so small that the planet mass dominated its of both planets from continuing to increase in the later stages of the local neighbourhood. Disc models with a range of masses corre- orbital evolution has been stressed by Crida et al. (2008). In addition, sponding to a range of accretion rates were considered, the smallest Murray, Paskowitz & Holman (2002) indicate that a residual disc corresponding to the late stages of the protoplanetary disc lifetime. of planetesimals could produce a similar effect. Simulations were run for up to 20 000 initial orbits of the inner We undertook 3D simulations that incorporated consideration planet. of the vertical structure of the disc. Both models that adopted a viscosity that was independent of θ and layered models for which a viscosity was only applied in the upper portion of the θ domain 5.1 Maintenance of a 2:1 commensurability were considered. The orbital evolution that was obtained was found When the mass ratio for both planets was 10−3, a 2:1 commen- to be in good agreement with that obtained from corresponding 2D surability was maintained for small enough disc masses. For our simulations. One effect seen in the 3D simulations that cannot be standard case with a low-mass disc and a corresponding steady- recovered from the 2D simulations is the vertical flow towards the state accretion rate of 6 × 10−10 M yr−1, the inward migration mid-plane in the interior neighbourhood of the planet. However, time −r/r˙ ∼ 1.5 × 105 inner planet initial orbital periods is a char- because this occurs in the gap region where the density is very low, acteristic viscous time-scale and so corresponds to standard type II this does not lead to significant departures from the 2D results for Downloaded from migration. the orbital evolution. Noting that observed parameters are somewhat uncertain, those In order to consider a system resembling the HD 6805 sys- for this model may approximately correspond to those for HD tem, we performed a simulation identical to the standard one 155358 and 24 Sextantis (see Table 1) if the inner planet semi- except that the mass of the outer planet was increased by a

major axis is respectively taken to be 0.64 and 1.33 au, respectively. factor of 2. This behaved like the standard case with mainte- http://mnras.oxfordjournals.org/ The inward migration times are then respectively ∼ 0.77 × 105 nance of a 2:1 commensurability and an inward migration rate and 2.3 × 105 yr which are relatively short compared to a charac- −r/r˙ ∼ 105 inner planet initial orbital periods. Using the same teristic protoplanetary disc lifetime. For illustrative purposes, let us scaling argument as above, the inner planet can be estimated to assume the planets have migrated in resonance for a time, t,and start at a radius being ∼5.5 au if resonant migration is assumed for use the scaling procedure given in Section 3.2 to estimate the ini- t = 106 yr. tial radius of the inner planet. For either of the two examples, we obtain r = (t/(1.5 × 105 yr))2/3 au ∼ 3.5 au for t = 106 yr. We note that this is beyond the ice line, with location estimated at about 2.7 5.2 Increasing planet mass and the formation au from a solar mass star (see e.g. Martin & Livio 2013). But we of a 3:1 resonance at University of Cambridge on August 10, 2016 emphasize that the scaling used restricts the disc model such that We have also studied a case with planet masses chosen to corre- ∝ r−2. While this might be relaxed to some extent, the surface spond to the HD 60532 system which has the larger planet mass density cannot exceed this projection by a large amount because a ratios 2.2 × 10−3 and 5.2 × 10−3 (see Table 1). These planets are commensurability with period ratio closer to unity would be formed observed to be in a 3:1 resonance (Laskar & Correia 2009). In our (see below). simulation, the planets attained a 3:1 resonant configuration with However, we note that the quoted eccentricity for the inner planet −r/r˙ ∼ 6 × 104 initial inner orbits. For the purposes of an illustra- in HD 155358 is 0.17 ± 0.03 (Robertson et al. 2012a). The mean tive discussion, if we assume that the scaling to larger radii discussed value is exceeded for the standard run at 4000 orbits after going in Section 3.2 applies, we find that if the system arrived in its present into resonance with the implication that the planets could not have location after having undergone inward migration in resonance for been in resonance longer than this time. If this is the case, the 106 yr, it should have started with the inner planet at an orbital ra- above discussion would have to be modified to allow the planets dius of ∼ 7.4 au. For a shorter evolution time of 4 × 105 yr, the to migrate independently from larger radii before converging on corresponding starting location shifts to 4 au. However, note that to resonance close to their final locations. This is likely to need as the orbital configuration obtained in the simulation is like that to be considered for different possible exterior disc models and in observed, the two planets may have only spent a relatively small addition the planets may have built up their masses as they went time in resonance, comparable to our simulation run time of ∼ 6000 (see e.g. Tadeu dos Santos et al. 2015). These considerations are orbits. In that case, the planets could have migrated independently, beyond the scope of this paper. None the less, because the migration starting at initial radii that did not differ by a large factor on account rates for single planets and the resonantly coupled planets are in of the single-planet migration rates being comparable. If this is the general similar, the estimated starting radii would also be similar case, detection of the system in resonance would be unlikely. On for disc models that are similar to those we considered. But note the other hand, only one system of this kind is currently known. that the attained eccentricities depend on the eccentricity damping rates which depend on the details of the disc model (see Crida et al. 2008). For example, we found that for the same amount of relative 5.3 Effect of increasing disc mass resonant migration, the entirely inactive disc model led to smaller eccentricities while the 3D layered model led to larger eccentricities. When the surface density or equivalently the steady-state accretion Thus, it is important to note that there is uncertainty as to how long rate was increased, the character of the migration of the resonantly the planets could have been in resonance. In the same context, we coupled pairs of Jupiter mass planets changed. When it was in- comment that migration in the completely active disc model was creased by a factor of 5, the planets are found to enter a 5:3 res- slower by a factor of ∼1.6 compared to the standard case on account onance which became unstable after about 5000 orbits leading to of its lower mass, that being determined so as to maintain the same a planet–planet scattering, as was found by Lee et al. (2009)who steady-state accretion rate as in the standard case. Furthermore, the adopted an N-body approach. The unstable character of the 5:3

MNRAS 461, 4406–4418 (2016) 4418 Q. Andre´ and J. C. B. Papaloizou resonance makes the observed occurrence of such a resonance less de Val-Borro M., Artymowicz P., D’ Angelo G., Peplinski A., 2007, A&A, likely than a 2:1 resonance. 471, 1043 If pairs of planets formed at a few au and then migrated to their Emelyanenko V., 2012, Sol. Syst. Res., 46, 321 present locations with the inner planet being at around 1 au (such Gammie C. F., 1996, ApJ, 457, 355 as for the HD 155358 and 24 Sextantis systems) while maintaining Ivanov P. B., Papaloizou J. C. B., Polnarev A. G., 1999, MNRAS, 307, 79 Johnson J. A. et al., 2011, AJ, 141, 16 a 2:1 commensurability for a characteristic time comparable to the Kley W., 1999, A&A, 208, 98 disc lifetime, the disc should have a low mass as might occur during Kley W., Lee M. H., Murray N., Peale S. J., 2005, A&A, 437, 727 the later stages of a protoplanatary disc lifetime. We have found Kley W., Bitsch B., Klahr H., 2009, A&A, 506, 971 that a disc with significantly larger mass produces an unstable 5:3 Laskar J., Correia A. C. M., 2009, A&A, 596, 5 resonance resulting in its observed occurrence being less likely. Latter H. N., Balbus S., 2012, MNRAS, 424, 1977 Although 3:2 resonances may be produced in other situations (e.g. Lee M. H., Peale S. J., 2002, ApJ, 567, 596 Rein et al. 2010, and see the end of Section 4.3 above), characteristic Lee A. T., Thommes E. W., Rasio F. A., 2009, ApJ, 691, 1684 evolution times are again short. Lin D. N. C., Papaloizou J. C. B., 1986, ApJ, 309, 846 Finally, consideration of systems containing a pair of giant planets Lin D. N. C., Papaloizou J. C. B., 1993, in Levy E. H., Lunine J. I., eds, with the innermost one being significantly more massive, for which Protostars and Planets III, p. 749 Martin R. G., Livio M., 2013, MNRAS, 428, L11 the mechanism outlined by Masset & Snellgrove (2001) may operate

Masset F. S., 2000, A&A, 141, 165 Downloaded from more efficiently, should be undertaken but is beyond the scope of Masset F., Snellgrove M., 2001, MNRAS, 320, L55 this paper. A recently discovered system of this kind is HD 204313 Mignone A., Bodo G., Massaglia S., Matsakos T., Tesileanu O., Zanni C., (Robertson et al. 2012b). This contains an inner planet of mass Ferrari A., 2007, ApJS, 170, 228 3.55 MJ with semi-major axis 3.04 au and an outer planet of mass Mignone A., Flock M., Stute M., Kolb S. M., Muscianisi G., 2012, A&A, 1.68 MJ with semi-major axis 3.93 au, the pair being in or close to 545, A152 a 3:2 commensurability. Accordingly, this will be the focus of a Mihalas D., Weibel Mihalas B., 1984, Foundations of Radiation Hydrody- http://mnras.oxfordjournals.org/ future study. namics. Oxford Univ. Press, New York Murray N., Paskowitz M., Holman M., 2002, ApJ, 565, 608 Nelson R. P., Papaloizou J. C. B., Masset F., Kley W., 2000, MNRAS, 318, ACKNOWLEDGEMENTS 18 Nelson R. P., Papaloizou J. C. B., 2002, MNRAS, 333, 26 We thank the anonymous referee for useful comments and sugges- Nelson R. P., Gressel O., Umurhan O. M., 2013, MNRAS, 435, 2610 tions which greatly enhanced the quality of this paper. QA was Pierens A., Nelson R. P., 2010, A&A, 520, id.A14 supported by ENS Cachan and thanks warmly DAMTP, University Rein H., Papaloizou J. C. B., Kley W., 2010, A&A, 510, A4 of Cambridge for their hospitality. Papaloizou J. C. B., 2003, Celest. Mech. Dyn. Astron., 87, 53 Rein H., Payne M. J., Veras D., Ford E. B., 2012, MNRAS, 426, 187 at University of Cambridge on August 10, 2016 Robertson P. et al., 2012a, ApJ, 749, 39 REFERENCES Robertson P. et al., 2012b, ApJ, 754, 50 Balbus S. A., Hawley J. F., 1991, ApJ, 376, 214 Sandor´ Z., Kley W., 2010, A&A, 517, 31 Baruteau C., Papaloizou J. C. B., 2013, ApJ, 778, 7 Shakura N. I., Sunyaev R. A., 1973, A&A, 24, 337 Baruteau C. et al., 2014, Protostars and Planets VI. Univ. Arizona Press, Snellgrove M. D., Papaloizou J. C. B., Nelson R. P., 2001, A&A, 374, 1092 Tucson, p. 667 Tadeu dos Santos M., Correa-Otto J. A., Michtchenko T. A., Ferraz-Mello Beauge C., Ferraz-Mello S., Michtchenko T. A., 2012, Res. Astron. Astro- S., 2015, A&A, 573, A94 phys., 12, 1044 Trifonov T., Reffert S., Tan X., Lee M. H., Quirrenbach A., 2014, A&A, Bryden G., Chen X., Lin D. N. C., Nelson R. P., Papaloizou J. C. B., 1999, 568, A64 ApJ, 514, 344 Uribe A. L., Klahr H., Henning T., 2013, ApJ, 769, 97 Calvet N., Muzerolle J., Briceno˜ C., Hernandez J., Hartmann L., Saucedo Wright J. T. et al., 2011, ApJ, 730, 93 J. L., Gordon K. D., 2004, AJ, 128, 1294 Correia A. C. M. et al., 2009, A&A, 496, 521 Crida A., Sandor´ Z., Kley W., 2008, A&A, 483, 325 Crida A., Baruteau C., Kley W., Masset F., 2009a, A&A, 502, 679 Crida A., Masset F., Morbidelli A., 2009b, ApJ, 705, 148 This paper has been typeset from a TEX/LATEX file prepared by the author.

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